A Coalescence Mechanism for the Coarsening Behavior ofPolymer Blends during a Quiescent Annealing Process.II. Polydispersed Particle System

WEI YU,1 CHIXING ZHOU,1 TAKASHI INOUE2

1 Department of Polymer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

2 Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Tokyo 152, Japan

Received 20 September 1999; revised 3 April 2000; accepted 19 June 2000

ABSTRACT: A coalescence mechanism for the coarsening behavior of polymer blendsduring a quiescent annealing process is proposed, in which the effect of differentparticle sizes is taken into consideration. All existing forces, such as thermal agitation,van der Waals attraction, interfacial tension, and viscous resistance, control the coars-ening process. The evolution of the particle size and its distribution are modeled witha particle population balance equation. The experimental investigation has been madewith the transmission electron microscopy observation of a polystyrene/poly(methylmethacrylate) blend during the annealing process. The model predictions are in goodagreement with our experiment and other authors’ work. The relative effects of systemparameters such as volume fraction, interfacial tension, and viscosity on the particlesize distribution are also discussed. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys38: 2390–2399, 2000Keywords: polymer blends; coalescence; annealing; polydisperse

INTRODUCTION

As we know, the performance of immiscible poly-mer blends with a particulate phase dependslargely on the mean particle size and size distri-bution. The morphology of polymer blendschanges greatly in two different stages: the pro-cessing and postprocessing stages. Although themorphology is mainly determined in processing, apostprocessing stage such as annealing can in-crease the particle size and influence the particlesize distribution. This coarsening behavior hasformerly been described by the Ostwald ripening(evaporation and condensation) mechanism [Lif-shitz–Slyozov–Wagner (LSW) mechanism]1,2 and

the collision–coalescence mechanism [Binder-Siggia (BS) mechanism].3,4

In the theory of Ostwald ripening, mass trans-fers between particles of different curvaturesthrough their surrounding continuous medium.The concentration of the dispersed-phase mate-rial at the surface of a particle is inversely relatedto the radius of curvature. Therefore, a smallparticle has a higher surface concentration incomparison with a large particle; this gives rise toa concentration gradient of disperse material inthe continuous phase. Then, the mass transferfrom the small particle to the large particlecauses small particles to be smaller and largeparticles to be larger. The LSW mechanism in-volves the solution of three equations:5 (1) a ki-netic equation gives the growth rate of an indi-vidual particle, (2) a continuity equation controlsthe particle size distribution, and (3) a conserva-tion equation keeps the quantity of materials in

Correspondence to: C. Zhou (E-mail: cxzhou@mail.sjtu.edu.cn)Journal of Polymer Science: Part B: Polymer Physics, Vol. 38, 2390–2399 (2000)© 2000 John Wiley & Sons, Inc.

2390

the system constant. The main results are thatthe average particle radius grows as t1/3 and thearbitrary distribution of particles will assume aspecific time-independent form.

The collision–coalescence theory considers theinfluence of Brownian motion on the coarsening ofdispersed particles. In the BS mechanism, parti-cle collisions are assumed to occur as a result ofthermal Brownian motion. The rate law R ; t1/3 isalso found, and a self-preserving distribution ispredicted.

Fortelny et al.6–9 considered another coales-cence mechanism. This course of coalescence ofdispersed particles can be divided into four stages:the particle approaching, the film draining, thefilm rupturing, and the neck relaxing. They tookthe film-drainage time as the coalescence timebetween two particles. They arrived at differenttime dependencies of the mean particle size byconsidering different interfacial mobilities anddriving forces. The exponent of the rate law R ; tn

varies from 15 to 1 under different interfacial mo-

bilities and driving forces. This mechanism pre-dicts substantially quicker coalescence than theone determined experimentally. Moreover, thereare some parameters assumed ad hoc that mayinfluence the predictions. Fortelny does notpresent a particle size distribution correspondingto this mechanism.

In part I of this series,10 we put forward a newcoalescence model in which interactions such asBrownian motion, van der Waals attraction, in-terfacial tension, and viscous resistance are alltaken into consideration. Moreover, unlike For-telny6–9, we take the addition of the draining timeof the matrix film and the merging time of dumb-bell-like particles as the coalescence time. Theresults show good consistency with experimentaldata.

In this article, we focus our attention on theparticle size polydispersity because the final prop-erties of blends depend not only on the meanparticle size but also on the particle size distribu-tion. We get the change in the particle size distri-bution with time as well as the evolution of themean particle size with time by solving the pop-ulation balance equations. Furthermore, we in-vestigate the coarsening behavior of polystyrene(PS)/poly(methyl methacrylate) (PMMA) blendsat 190 °C. The predictions of the particle sizedistribution are in agreement with the experi-ments.

THEORY

The coalescence process of two particles of differ-ent sizes can be divided into four stages (Fig. 1):the approach of the particles and the formation ofa thin film between two particles, the drainage ofthe matrix between two particles, the rupture ofthe film, and the evolution of dumbbell-like par-ticles into spheres. The details of each stage aredescribed in part I of this series10 and are notlisted here. The total coalescence time, t, of twoparticles is the summation of the matrix drainagetime, tdrain, and the merge time of the dumbbellparticle, tmerge:

t 5 tdrain 1 tmerge (1)

The film rupture time is ignored because it isconsidered a much quicker stage. The approachtime of two particles is not included because thisstage does not really change the particle size. Itjust changes the spatial distribution and therelative position of the particles. Hence, it will

Figure 1. Schematic diagram of the coarsening process: (I) the approach of theparticles and the formation of the matrix film, (II) the drainage of the matrix film, (III)the rupture of the matrix film, and (IV) the merge of the dumbbell-like particles.

POLYDISPERSED PARTICLE SYSTEM 2391

influence the coalescence probability, which isstrongly dependent on the relative particle posi-tion and particle size.

Coalescence Time

The drainage time can be obtained by solving thefollowing equation:10

dhr

dt 5 ahr3 1 bhr 1 d (2)

with

a 5G

6hmReqRf2 , b 5 2

12Rf

­

­r @r~u1 1 u2!#r5Rf ,

d 5A

18phmRf2 (3)

where hm is the matrix viscosity; Req is the equiv-alent radius; Rf is the radius of the film, which isassumed to be time-independent;11 hr is the filmthickness at the rim of film; and u1 and u2 are thetangential velocities on the particle surface,which can be expressed by a boundary integralformula12,13 (see part I of this series10 for details).

The merge time of two particles can be writtenas follows:10

tmerge 5Reqh

2Î32G(4)

This equation was originally derived for themerge process of two particles of equal sizes. Fordifferent particle sizes, we use the equivalent par-ticle radius, Req, instead of the particle radius, R.After the drainage time and merge time are cal-culated, the total coalescence time can be calcu-lated with eq 1.

Population Balance Equation

The distribution of the drop sizes of annealingimmiscible polymer blends is controlled by coales-cence phenomena. This process contributes to therate of the appearance (birth) and disappearance(death) of particles. Particles with a given vol-ume, Vi, can appear as a result of the coalescenceof two smaller particles with volume Vi 2 Vj andVj, respectively (Vj , Vi). Here, the subscripts iand j denote particles with different volumes. The

death of a particle can happen because of thecoalescence of this particle with others.

The population balance equation representsthe conservation of the particle volume during thecoalescence process in terms of the rate of changein the number particle density, n(Vi,t), for eachparticle size at an arbitrary instant, t. This can bedetermined by the two opposing rates of particlebirth, B(Vi,t), and death, D(Vi,t). The populationbalance can be written as follows:14

dn~Vi, t!dt 5 B~Vi, t! 2 D~Vi, t! (5)

The rates of particle birth and death depend onthe number particle density. The more a certainsized particle has, the more likely it will coalescewith others. The rates of particle birth and deathare also related to the efficiency of coalescence.The change of the number particle density will bemuch faster if the coalescence time is shorter.Therefore, B(Vi,t) and D(Vi,t) can be written asfollows:

B~Vi, t! 5 Oj51

@i/2#

n~Vi 2 Vj, t!f~Vi 2 Vj, Vj!

3 p~Vi 2 Vj, Vj!n~Vj, t! (6)

D~Vi, t! 5 Oj51

N

n~Vi, t!f~Vi, Vj!

3 p~Vi, Vj!n~Vj, t! (7)

where p is the probability that the surface of acertain particle is within a suitable range fromthe surface of any other particle and N is the totalnumber of different particle sizes. In eq 6, theupper limit of summation, [i/2], means the maxi-mum integer is less than i/2. f(Vi,Vj) denotes therate of change in the particle volume per unit oftime:

f~Vi, Vj! 5Vi 1 Vj

2t~Vi, Vj!(8)

t(Vi,Vj) is the total coalescence time of particleswith volumes Vi and Vj. Additionally, the numberdensity of particles of different sizes must satisfythe condition of conservation of the overall parti-cle concentration:

2392 YU, ZHOU, AND INOUE

Oi51

N

Vin~Vi, t! 5 f (9)

where f is the volume fraction of the dispersedphase. The particle number of different particlesizes can be solved from eqs 5–8 with a fourth-order Runge–Kutta formula. With the particlesize distribution known, several mean particleradii can be calculated with normalized particlesize distribution:

fiN 5

n~Vi!Oi

n~Vi!~Normalized distribution!

Rn 5 Oi

fiNRi ~Number-average radius!

Rv 5 Î3 Oi

fiNRj

3 ~Volume-average radius!

Rw 5

Oi

fiNRi

2

Oi

fiNRi

~Weight-average radius!

PI 5 Rv/Rn ~Polydispersity index! (10a,b,c,d,e)

Here we use PI to describe the polydispersity ofthe particle size because Rv 5 Rw 5 Rn for mono-dispersed distribution. From eq 5 together witheqs 6 and 7, we can easily get the equation for amonodispersed system:

dNdt 5 2N z f z p z N (11)

where the particle birth rate is set to be 0 becausethere is no smaller particle to coalesce.

Touching Particle Probability

For multibody systems, polydispersity leads to awider choice of possible definitions for nearest-neighbor functions. There are essentially two dif-ferent kinds of nearest-neighbor functions forthree-dimensional spherical particle systems: onespecifying the nearest sphere surface to a refer-ence point and the other specifying the nearestsphere center to a reference point. For a monodis-persed system, these two nearest-neighbor func-

tions contain the same function. For polydis-persed systems, the nearest-surface distributionfunctions and nearest-center distribution func-tions are generally distinctly different. Lu andTorquato15,16 developed a formalism to calculatethe nearest-surface distribution function,Hp(r)dr, which means the probability of findingthe nearest particle surface between r and r 1 dr,given any three-dimensional sphere of radius Riat some arbitrary position in the system:

Hp~r! 5 pr~c 1 2 dr 1 3gr2!Ep~r! (12a)

Ep~r! 5 exp$2pr@c~r 2 R!

1 d~r2 2 R2! 1 g~r3 2 R3!# (12b)

where r is the distance of the nearest surface tothe center of a reference particle and

c 54^R2&

1 2 f, d 5

4^R&

1 2 f1

12j2

~1 2 f!2 ^R2&,

g 54

3~1 2 f!1

8j2^R&

~1 2 f!2 116j2

2A3~1 2 f!3 ^R2& (13)

jk 5p

3 r2k21^Rk&

^A~R!& 5 E0

`

A~R!f N~R! dR

where r is the total particle number, A 5 2 is theCarnahan–Starling approximation, A 5 3 is thescaled-particle approximation, and A 5 0 is thePercus–Yevick approximation. Equation 12 givesthe nearest-surface distribution, which can bemodified to be the nearest-center distribution:

hp~r! 5 pr@c 1 2d~r 2 R2!

1 3g~r 2 R2!2#ep~r! (14a)

ep~r! 5 exp$2pr@c~r 2 R2 2 R1!

1 d~~r 2 R2!2 2 R1

2! 1 g~~r 2 R2!3 2 R1

3!# (14b)

where r is the distance of the nearest particlecenter to the center of a reference particle.

Therefore, the touching particle probability, p,can be expressed as follows:

POLYDISPERSED PARTICLE SYSTEM 2393

p 5 Ex0

x1

hp~r! dr (15)

The upper and lower bounds of the integrate canbe defined as follows:

x0 5 R1 1 R2 1 h0, x1

5 R1 1 R2 1 h0 1 Dh1 1 Dh2 (16)

where Dhi is the translational distance by Brown-ian motion, which can be estimated by

Dhi 5 Î2Ddit (17)

where Ddi is the diffusion coefficient approxi-mated by the Einstein equation:17

Ddi 5 kBT/zi (18)

where zi is a friction constant, which is given as

zi 5 6phmRi, i 5 1, 2 (19)

Equation 16 means that only when the distancebetween two particle centers is within the upperand lower bounds can two particles touch eachother. Brownian motion promotes the touch prob-ability of two particles.

Now we know all the parameters needed forsolving the population balance equations (eqs5–8). To calculate the change of the number den-sity distribution with time, the initial numberdensity distribution should be known. The initialdistribution can be obtained from experimentaldata, or a certain predefined distribution, such asmonodistribution, normal distribution, or lognor-mal distribution, simply can be assumed. In thisarticle, we use an experimental initial distribu-tion in comparing the model with the experimentand use lognormal distribution in discussing theinfluencing factor of the particle size distribution.Lognormal distribution has the following form:

y 5 y0 1 Ae2ln2~x/xc!/2w2 (20)

where xc can be regarded as a relative average ofdistribution and w is the relative dispersion ofdistribution. In experiments, some other polydis-persity indices are used, such as Rv/Rn or Rw/Rn.Later, we show that either w or Rv/Rn has the

same information about polydispersity and xc hastrend similar to that of Rn or Rv.

Equation 5, a population balance equation, is afirst-order, nonlinear, ordinary differential equa-tion set; it can be solved by the Runge–Kuttamethod easily if the number of equations is fixed.Because these equations are used to describe thechange in the particle number in multibody sys-tems, difficulty comes from the variation of equa-tions. In fact, the equations of small particles areof no use when small particles have completelydisappeared; the equations of large particle areadded when large particles appear as the result ofthe coalescence of small particles. For example, ifinitially there are particles with radii rangingfrom 1 to 5 mm, the number of particles withdifferent volumes that need to be treated is 125.In a single time step, these 125 different particlesmay coalescence to form, at most, 250 differentparticles. Hence, the number of different particlesincreases in a geometric series. It is very difficultto take into account all the particles during thesimulation of a long-time annealing process. Toeconomize computing time yet not influence theaccuracy of the solution, we ignore those particleswhose population is only a very small portion ofall particles, and most of these particles are largeones. Although such a method may cause somedeviation, it greatly decreases the scale of differ-ential equations and shortens the computationtime.

EXPERIMENTAL

The polymers used in this study are PS [PS666D;Dow Chemical; weight-average molecular weight(Mw) 5 200,000] and PMMA (Rohm & Haas; Mw5 111,000). PS/PMMA (85/15 weight ratio) blendswere prepared in a Haake Rheomix 600 batchmixer at a rotor speed of 50 rpm at 200 °C for 7min. We chose transmission electron microscopy(TEM) as the characterization technique becausewe can only obtain the mean particle size by time-resolved light scattering. For TEM observation ofthe coarsening behavior, the PS/PMMA blend wasannealed in a hot chamber at 190 °C for 2 h, 5 h,11 h, and 14 h and then quenched in iced water.The annealed and quenched specimens were em-bedded in epoxy resin. After the epoxy was cured,the embedded specimens were microtomed intoultrathin sections (;50 nm thick) at room tem-perature with an ultramicrotome (Reichert Ul-tracut N) with a diamond knife. To provide con-

2394 YU, ZHOU, AND INOUE

trast, the microtomed sections on the grid wereexposed to the vapor of ruthenium tetraoxide,which preferentially stains PS. The phase struc-ture was observed by TEM (JEOL 100CX) at anaccelerating voltage of 100 keV. Figure 2 showsthe TEM micrographs of a PS/PMMA blend at

190 °C for t 5 0, 5, and 11 h. Five micrographswere taken for each specimen. Each micrographcontained about 200 particles. The area of theindividual particle, Si, was directly determinedwith image analysis software (Scion Image),which measures the domain area. The diameter ofdispersed particles, Di, was calculated with thedomain shape assumed to be circular:

Di 5 2ÎSi

p(21)

The zero-shear-rate viscosities of PS and PMMAat different temperatures were measured with aRheometrics dynamic spectrometer (model RDS-7700). The zero-shear viscosity of PS was9.433103 Pa z s, and the zero-shear viscosity ofPMMA was 2.873105 Pa z s at 190 °C.

RESULTS AND DISCUSSION

We first simulated the coarsening behavior withthe population balance equations. Some parame-ters needed for computation are the volume frac-tion, f 5 0.131 (with the density values, r, atroom temperature: rPS 5 1.04 and rPMMA 5 1.20),and the interfacial tension, G 5 1.1 3 1023 N/m at180 °C (dG/dt 5 20.013 3 1023 N/m). The initialdistribution of the 15/85 PS/PMMA blend wastaken from the experiment. The time variation ofthe average radius of dispersed particles is shownin Figure 3. The model predictions (solid lines)agree with the experimental data. Rn grows

Figure 2. TEM micrographs of the 85/15 PS/PMMAblends: (a) as mixed, (b) after annealing for 5 h, and (c)after annealing for 11 h at 190 °C.

Figure 3. Time variation of the number-average andvolume-average radius of the 85/15 PS/PMMA blend at190 °C. The solid lines represent the model predictions,and the scattered dots represent experimental datafrom TEM. The dashed line represents the time varia-tion of the polydispersity (Rv/Rn).

POLYDISPERSED PARTICLE SYSTEM 2395

nearly linearly with annealing, whereas Rv growswith the rate law exponent about 0.6, which islarger than the traditional value, 1

3. The polydis-persed model does not predict the growth of theaverage particle size very well. The polydispersityindex, Rv/Rn, increases from the initial value 1.12to the final value 1.3. Experimental results showanother tendency of Rv/Rn, to first increase andthen decrease. However, both experiments andmodel predictions for Rv/Rn vary within a smallinterval, from which we cannot conclude whetherthe distribution state is broader or narrower. Thereason for the discrepancy lies in many aspects.One comes from the solution of the model. As wehave stated previously, we ignore some particlesin the computation that constitute only a verysmall portion of all the particles. Although thismay be regarded as appropriate in mathematics,these ignored particles will influence the coales-cence of other particles and cause a shift in thepredicted distribution. Consequently, the meanparticle size predicted might depart slightly fromthat observed in the experiment. Another reasonis related to the statistics of the dispersed parti-cles. Because the number of TEM photographs isprobably not enough to represent all the particles,it shows only a small portion of all the particles,although it is representative. Both factors mayintroduce differences between the experimentsand model predictions.

The model predictions of the time variation ofthe particle size distribution are shown in Figure4. We find that the predictions agree well with theexperimental distribution, except at the late stage

of annealing. We can see clearly from Figure 4that the difference between the model predictionand experiment at 14 h is the position of thecenter of distribution. As previously stated, weomit those particles that only take a very smallportion of all the particles, although the absolutenumber of such particles is not small. Omittingthose particles in the computation might cause aslow growth of large particles, which may be thesource of the discrepancy in the last graph inFigure 4.

We also compare our model prediction withother authors’ experiments. Figure 5(a) shows thenumber-average radius of a polycarbonate (PC)/polypropylene (PP) blend system with 23% PC at250 °C.18 The zero-shear viscosities for PC and PPare 2900 and 920 Pa z s, respectively. The cumu-lative percentage of the particle size is shown inFigure 5(b). The initial particle size distributionis obtained from curve 1 (scattered dots) in Figure

Figure 4. Particle size distribution of the 85/15 PS/PMMA blend at different annealing times under 190°C. The vertical bars represent experimental resultsfrom TEM, and the solid lines represent model predic-tions of the particle size distribution.

Figure 5. Cumulative distribution of the particle sizeof the PC/PP (23% PC) blend annealing at 250 °C atdifferent annealing times: (a) mean particle size and (b)particle size distribution. The solid lines are computa-tional results, and the scattered dots are experimentalresults.

2396 YU, ZHOU, AND INOUE

5(b). Both the mean particle size and particle sizecumulative distribution predicted by our modelagree with the experiment.

To find out how system parameters, such as thevolume fraction of the disperse phase, interfacialtension, matrix, and disperse phase zero-shearviscosity affect the particle size and its distribu-tion and, more importantly, whether the particlesize distribution changes during annealing, weassume the particle disperses with a specific dis-tribution form, lognormal distribution19. The fol-lowing discussion only applies to the lognormaldistribution. We first check the influence of thevolume fraction. Figure 6 shows the influence ofthe volume fraction on the particle size and itsdistribution. We find that the particle size distri-bution changes with the annealing time [Fig. 6(a)shows an example of f 5 0.4, whose morphologyis assumed to be particulate]. However, the dis-tributive form does not change, which means theparticle size distribution still obeys the preas-sumed lognormal distribution but with differentparameters. Thus, we can use model parameters(xc and w) to study the change in the size distri-bution. From Figure 6(a), perhaps one would saythat the size distribution becomes broad. In fact,we find that both the model parameter w and thepolydispersity index Rv/Rn decrease with anneal-ing time [Fig. 6(c)], which means the size distri-bution does not become broader but narrower.Although w and Rv/Rn are not exactly equal, theyshow the same trend. Therefore, both can be usedto describe the polydispersity for lognormal dis-tribution. Moreover, a small volume fraction haslittle influence on it, and a large volume fractioncauses w and Rv/Rn to decrease even more withtime. The larger the volume fraction is, the moresignificant the decrease of w and Rv/Rn is duringthe early stage of annealing. In the late stage ofannealing, w and Rv/Rn curves tend toward equi-librium. The time variations of the number-aver-age radius and model parameter xc are shown inFigure 6(b). Similarly, both xc and Rn can be usedto describe the particle size of lognormal distribu-tion, although they are not exactly same. Theresults show that a large volume fraction causesthe particle size to increase more rapidly. More-over, the rate law exponent decreases from nearly1 for f 5 0.01 to 0.45 for f 5 0.4. This is inaccordance with the predictions using a monodis-persed model [see Fig. 5(a) in part I of this se-ries10). The narrowing distribution agrees withthe experiments done by Favis.18 However, not allkinds of distribution become narrower during an

annealing process. The distributions in our exper-iment and Jang et al.’s20 experiment are not log-normal, and the results show a nonnarrowing ofdistribution. What causes such a discrepancy be-

Figure 6. Influence of the disperse volume fractionon the evolution of the particle size and its distribution:(a) time variation of the particle size distribution at f5 0.4, (b) time variation of the number-average radius(solid lines) and xc (scattered dots) at different volumefractions, and (c) time variation of the polydispersity,Rv/Rn (solid lines), and w (scattered dots) at differentvolume fractions.

POLYDISPERSED PARTICLE SYSTEM 2397

tween lognormal and nonlognormal distributionsis still unclear.

Similarly, we check the influence of interfacialtension and matrix zero-shear viscosity and showthe results in Figures 7 and 8, respectively. It canbe seen from Figure 7(a) that large interfacialtension causes the particle size to increase morerapidly, and the rate law exponent becomessmaller, from nearly 1 for G 5 0.0005 N/m to 0.6for G 5 0.01 N/m. Low interfacial tension makesthe particle size increase linearly, whereas withhigh interfacial tension there is an apparent slow-down in the late stage of annealing. Of course, theproportionality constant is not identical for differ-ent interfacial tensions, volume fractions, and vis-cosities. That means both the proportionality con-stant and the rate law exponents are affected bythe interfacial tension, as well as by the volumefraction and viscosity. In the late stage of anneal-

ing, the system has more large particles than inthe initial state. Because the Brownian mobilityof large particles is weaker than that of smallparticles and the average distance between twoneighboring particles increases with particle size,the probability of particle coalescence is lowered.Therefore, the increase of large particles slowsdown, which results in the slowdown of thegrowth of the mean particle size. However, theslowdown degree is different for different sys-tems. The systems with high interfacial tensions,large volume fractions, and small viscosities willshow slowdowns. This is because those factorswill increase the variation of particle numbers perunit of time and amplify the influence of largeparticles. In Figure 8, the prediction is just likethe previous analysis: a smaller matrix zero-shear viscosity makes the particle size increasemore quickly and the polydispersity index de-crease more quickly.

Figure 7. Influence of the interfacial tension on theevolution of the particle size and its distribution: (a)time variation of the number-average radius (solidlines) and xc (scattered dots) at different interfacialtensions and (b) time variation of the polydispersity,Rv/Rn (solid lines), and w (scattered dots) at differentinterfacial tensions.

Figure 8. Influence of the matrix zero-shear viscosityon the evolution of the particle size and its distribution:(a) time variation of the number-average radius (solidlines) and xc (scattered dots) at different matrix zeroviscosities and (b) time variation of the polydispersity,Rv/Rn (solid lines), and w (scattered dots) at differentmatrix zero viscosities.

2398 YU, ZHOU, AND INOUE

By checking the influence of the system param-eters on the polydispersity, we find that the poly-dispersity index decreases if the initial size dis-tribution satisfies lognormal distribution, whichis supported by Favis’s18 results. This is not con-sistent with the previous prediction, in which theinitial size distribution was taken from our exper-iment. This inconsistency suggests that differentinitial distributions might cause different evolu-tions of the polydispersity index.

CONCLUSION

We have simulated the coarsening behavior of apolydispersed system by combining populationbalance equations and the coalescence mecha-nism put forward in part I of this series.10 Themodel prediction agrees well with our experimen-tal results and is consistent with other publishedexperimental data within a tolerable margin oferror. The exponent a of the mean-radius–timepower rate relation R ; kta decreases, whereasthe proportionality constant k increases for asmaller matrix zero-shear viscosity, higher inter-facial tension, and large volume fraction. Thismatches the results from the monodispersedmodel. For systems initially having lognormaldistribution, we find the form of distribution doesnot change and the polydispersity index tends todecrease during annealing.

This work was supported by research grants from theNational Natural Science Foundation of China and theEno Science Foundation of Japan.

REFERENCES AND NOTES

1. Lifshitz, I. M.; Slyozov, V. V. J Phys Chem Solids1961, 19, 35.

2. Wagner, C. Z. Elektrochemie 1961, 65, 581.3. Binder, K.; Stauffer, D. Phys Rev Lett 1974, 33,

1006.4. Siggia, E. D. Phys Rev A 1979, 20, 595.5. Cavanaugh, T. J.; Nauman, E. B. J Polym Sci Part

B: Polym Phys 1998, 36, 2196.6. Fortelny, I.; Kovar, J. Polym Compos 1988, 9,

119.7. Fortelny, I.; Zivny, A. Polymer 1995, 36, 4113.8. Fortelny, I.; Zivny, A. Polymer 1998, 39, 2669.9. Fortelny, I.; Zivny, A.; Juza, J. J Polym Sci Part B:

Polym Phys 1999, 37, 181.10. Yu, W.; Zhou, C.; Inoue, T. J Polym Sci Part B:

Polym Phys 2000, 38.11. Ivanov, I. B.; Dimitrov, D. S. In Thin Liquid Films;

Ivanov, I. B., Ed.; Marcel Dekker: New York, 1988;pp 379–497.

12. Barnocky, G.; Davis, R. H. Int J Multiphase Flow1989, 15, 627.

13. Davis, R. H.; Schonberg, J. A.; Rallison, J. M. PhysFluids A 1989, 1, 77.

14. Patlazhan, S. A.; Lindt, J. T. J Rheol 1996, 40,1095.

15. Lu, B.; Torquato, S. Phys Rev A 1992, 45, 5530.16. Lu, B.; Torquato, S. J Chem Phys 1993, 98, 6472.17. Doi, M.; Edwards, S. F. The Theory of Polymer

Dynamics; Oxford University Press: New York,1986; pp 46–83.

18. Favis, B. D. J Appl Polym Sci 1990, 39, 285.19. Wu, S. Polym Eng Sci 1987, 27, 335.20. Jang, B. Z.; Uhlmann, D. R.; Vander Sande, J. B.

Rubber Chem Technol 1983, 57, 291.

POLYDISPERSED PARTICLE SYSTEM 2399